No abstract

Let A be a normal operator in g ( H ) , H a complex Hilbert space, and let g A = 2 { A X -X A : X E g ( H ) } be the commutator subspace of B ( H ) associated with A . If B in g(H)commutes with A, then B is orthogonal to 9, with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in 9,. This was proved by J. ANDERSON in 1973 and extended by P. J. MAHER with respect to the Schatten p-norm recently.We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in W , and prove that the metric projection of H onto . %!A is continuous.

In a central paper on smoothness of best approximation in 1968 R. Holmes and B. Kripke proved among others that on ‫ޒ‬ n , endowed with the l-norm, 1p-ϱ, p the metric projection onto a given linear subspace is Lipschitz continuous where the Lipschitz constant depended on the parameter p. Using Hoffman's Error Bounds as a principal tool we prove uniform Lipschitz continuity of best l-app proximations. As a consequence, we reprove and prove, respectively, Lipschitz Ž. continuity of the strict best approximation sba, p s ϱ and of the natural best Ž. approximation nba, p s 1 .

No abstract

In this note we consider best approximation in 1-norm from subspaces of real n-dimensional spaces. Some discrete Kolmogorov-type criteria characterizing elements of best approximation are verified.2002 Elsevier Science (USA)